Problem 24
Question
Profit A company is increasing the production of a product at the rate of 25 units per week. The demand and cost functions for the product are given by \(p=50-0.01 x\) and \(C=4000+40 x-0.02 x^{2} .\) Find the rate of change of the profit with respect to time when the weekly sales are \(x=800\) units. Use a graphing utility to graph the profit function, and use the zoom and trace features of the graphing utility to verify your result.
Step-by-Step Solution
Verified Answer
The rate of change of the profit with respect to time when the weekly sales are 800 units is 26 units per week.
1Step 1: Determine the Revenue Function
The revenue is the product of the price and the quantity sold, \(R = p * x\). Here, \(p = 50 - 0.01x\). Therefore, \(R = x*(50 - 0.01x) = 50x - 0.01x^2\).
2Step 2: Determine the Profit Function
Profit (\(P\)) is revenue (\(R\)) minus cost (\(C\)). We already found \(R\) and we know \(C = 4000 + 40x - 0.02x^2\). Thus, \(P = R - C = 50x - 0.01x^2 - (4000 + 40x - 0.02x^2) = 10x + 0.01x^2 - 4000\).
3Step 3: Differentiate the Profit Function
Differentiating \(P\) with respect to \(x\), we get \(P' = dP/dx =10 + 0.02x\). This expression represents the rate of change of profit with respect to sales.
4Step 4: Substitute the Given Sales Quantity
We are required to find the rate of change of profit when \(x=800\) units. Substituting this into our derivative function obtained in step 3, \(P'(800) =10 + 0.02 * 800 = 26\). Hence the rate of change of profit with respect to time when the weekly sales are 800 units is 26 units per week.
5Step 5: Verify the Result using Graph
Plot the function \(P(x)=10x + 0.01x^2 - 4000\) using a graphing utility. Use the zoom and trace features to confirm that the slope of the function at \(x=800\) is indeed 26.
Key Concepts
Revenue Function CalculusProfit Function CalculusDifferentiation in EconomicsGraphical Analysis in Calculus
Revenue Function Calculus
The revenue function is at the heart of a business's success, representing the total amount of money generated from sales. It's crucial for students to understand how to model and analyze revenue using calculus.
The formula for the revenue function is quite simple: it's the product of the price per unit, denoted as p, and the quantity sold, x. This can be expressed as R(x) = p * x. In calculus, we often see revenue functions with price as a function of quantity, which accounts for the diminishing price with increased supply, commonly seen in economics.
When dealing with exercises like the one provided, we start by defining our revenue function based on the given price function. The revenue function, in this case, becomes a quadratic equation when we plug in the price per unit. A grasp of quadratic functions is essential here because they often arise in economic scenarios.
Once the revenue function is determined, we can use differentiation, another foundational calculus tool, to find the rate at which revenue is changing with respect to sales. This will tell us, essentially, how sensitive our revenue is to changes in the number of units sold.
The formula for the revenue function is quite simple: it's the product of the price per unit, denoted as p, and the quantity sold, x. This can be expressed as R(x) = p * x. In calculus, we often see revenue functions with price as a function of quantity, which accounts for the diminishing price with increased supply, commonly seen in economics.
When dealing with exercises like the one provided, we start by defining our revenue function based on the given price function. The revenue function, in this case, becomes a quadratic equation when we plug in the price per unit. A grasp of quadratic functions is essential here because they often arise in economic scenarios.
Once the revenue function is determined, we can use differentiation, another foundational calculus tool, to find the rate at which revenue is changing with respect to sales. This will tell us, essentially, how sensitive our revenue is to changes in the number of units sold.
Profit Function Calculus
Understanding the profit function is almost like having a financial compass for a business. It helps in pinpointing whether a business's strategy is sailing towards success or if it needs a course correction.
The profit function, denoted by P(x), is found by subtracting the cost function C(x) from the revenue function R(x), symbolized by P(x) = R(x) - C(x). As we've already established the revenue function, finding the profit function simply involves applying this formula. Remember, profit is what remains after all expenses are paid, so it's a critical measure of business performance.
In the given exercise, we demonstrate this process. After defining the revenue and cost functions, the profit function is derived. This eventually forms the basis of analyzing how much profit will change based on different sales volumes – a key question for any business.
The profit function, denoted by P(x), is found by subtracting the cost function C(x) from the revenue function R(x), symbolized by P(x) = R(x) - C(x). As we've already established the revenue function, finding the profit function simply involves applying this formula. Remember, profit is what remains after all expenses are paid, so it's a critical measure of business performance.
In the given exercise, we demonstrate this process. After defining the revenue and cost functions, the profit function is derived. This eventually forms the basis of analyzing how much profit will change based on different sales volumes – a key question for any business.
Differentiation in Economics
Economics and calculus are like two peas in a pod: they work hand in hand to analyze changes and trends in economic behaviors. Differentiation, a fundamental tool in calculus, is primarily used to examine rates of change.
For instance, when we differentiate a profit function P(x) with respect to the quantity x, like in our exercise, we get P'(x), which tells us how our profit is expected to change as we sell one additional unit. This marginal analysis method is a cornerstone of economic strategy and policy because understanding marginal changes provides powerful insights into overall economic trends.
In the context of the exercise, we differentiate the profit function to find the rate of change of profit with respect to sales (quantity sold), an essential analysis that can lead to informed decision making for pricing, production, and sales strategies.
For instance, when we differentiate a profit function P(x) with respect to the quantity x, like in our exercise, we get P'(x), which tells us how our profit is expected to change as we sell one additional unit. This marginal analysis method is a cornerstone of economic strategy and policy because understanding marginal changes provides powerful insights into overall economic trends.
In the context of the exercise, we differentiate the profit function to find the rate of change of profit with respect to sales (quantity sold), an essential analysis that can lead to informed decision making for pricing, production, and sales strategies.
Graphical Analysis in Calculus
A picture is worth a thousand words, and in calculus, a graph can be worth even more. Graphical analysis is a visual tool that helps to understand complex functions and their behaviors.
Using graphing utilities, we can plot the profit function and observe its shape, which can indicate how profit changes with different levels of production and sales. Features like zoom and trace allow us to focus on specific points and accurately determine the slope at those points.
The slope of the tangent to the profit function graph at any given point tells us the instantaneous rate of change of profit at that level of sales. In our exercise, by graphically analyzing the plot of P(x) at x = 800, we can visually confirm that the rate of change of profit is consistent with the numerical derivative we calculated. This visual confirmation is a powerful tool for students to understand the abstract concept of derivatives in a more concrete and approachable way.
Using graphing utilities, we can plot the profit function and observe its shape, which can indicate how profit changes with different levels of production and sales. Features like zoom and trace allow us to focus on specific points and accurately determine the slope at those points.
The slope of the tangent to the profit function graph at any given point tells us the instantaneous rate of change of profit at that level of sales. In our exercise, by graphically analyzing the plot of P(x) at x = 800, we can visually confirm that the rate of change of profit is consistent with the numerical derivative we calculated. This visual confirmation is a powerful tool for students to understand the abstract concept of derivatives in a more concrete and approachable way.
Other exercises in this chapter
Problem 24
Find the marginal revenue for producing units. (The revenue is measured in dollars.) $$ R=30 x-x^{2} $$
View solution Problem 24
Use Example 6 as a model to find the derivative. $$ y=\frac{2}{3 x^{2}} $$
View solution Problem 24
Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. Equation \(\quad\) Point \((x+y)^{3}=x^{3}+y^{3} \quad(-1,1)\)
View solution Problem 24
Use the General Power Rule to find the derivative of the function. $$ y=\left(2 x^{3}+1\right)^{2} $$
View solution